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\pub{2009}{1}{3}{1}
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\topic{Lecture 2.4 \\Matrix\\ \scriptsize Linear Dependence of Vectors (06 Nov 2009)}
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\section{Linear Dependence of Vectors}
The $n$ Vectors $X_1, X_2, ..., X_n$ of the same order, are said to be dependent if there exist $n$ scalars $\lambda_1, \lambda_2, ..., \lambda_n$ (not all zero) such that 
\begin{equation}
\lambda_1 X_1 + \lambda_2 X_2 + ... + \lambda_n X_n = O
\end{equation}
otherwise these are linearly independent.

\begin{example}
Examine the dependence of following vectors.
$X_1=(1, 2, 4)$, $X_2=(2, -1, 3)$, $X_3=(0, 1, 2)$, $X_4=(-3, 7, 2)$, 
\end{example}
\paragraph{Solution} 
Consider the matrix equation
\[\lambda_1 X_1 + \lambda_2 X_2 + \lambda_3 X_3 + \lambda_4 X_4 = O\]
\[\lambda_1 (1, 2, 4) + \lambda_2 (2, -1, 3)+ \lambda_3 (0, 1, 2) + \lambda_4 (-3, 7, 2) = O\]
This is the homogeneous equation systen and may be written as 

\[
\left[
\begin{array}[pos]{rrrr}
1 & 2 & 0 & -3  \\
2 & -1 & 1 & 7  \\
4 & 3 & 2 & 2  \\
\end{array}
\right]
\left[
\begin{array}[pos]{c}
\lambda_1 \\
\lambda_2 \\
\lambda_3 \\
\lambda_4 \\
\end{array}
\right]
=
\left[
\begin{array}[pos]{c}
0 \\ 0 \\ 0 \\
\end{array}
\right]
\]

Apply $R_2-2R_1$, $R_3-4R_1$, $R_3-R_2$, 

\[
\left[
\begin{array}[pos]{rrrr}
1 & 2 & 0 & -3  \\
0 & -5 & 1 & 13  \\
0 & 0 & 2 & 14  \\
\end{array}
\right]
\left[
\begin{array}[pos]{c}
\lambda_1 \\
\lambda_2 \\
\lambda_3 \\
\lambda_4 \\
\end{array}
\right]
=
\left[
\begin{array}[pos]{c}
0 \\ 0 \\ 0 \\
\end{array}
\right]
\]
\[
\lambda_1 + 2 \lambda_2 -3 \lambda_4 =0,~~~
-5 \lambda_2 + \lambda_3 + 13 \lambda_4 =0,~~~
\lambda_3 + \lambda_4 =0 \\
\]
Let $\lambda_4 = k$, for all $k$, then
\[\lambda_1 = \frac{-9}{5}k,~~~~\lambda_2 = \frac{12}{5}k,~~~~ \lambda_3 = -k,~~~~\] 
Hence the given vectors are dependent.
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\section{Problems}
\begin{enumerate}
\item  Examine the following vectors for linear dependence and find the relation if it exists.
$X_1 =(1,2,4)~~X_2 =(2,-1,3)~~X_3 =(0,1,2)~~X_4 =(-3,7,2) $

\item  Difine linear dependence and independence of vectors. Examination for linear dependence 
$[1,0,2,1],[3,1,2,1],[4,6,2,-4],[-6,0,-3,-4]$, and find the relation between them, if possible.

\item  Is the system of vectors $X_1=(2,2,1)^t,X_2=(1,3,1)^t,X_3=(1,2,2)^t$ linearly dependent?

\item  Examine the following system of vectors for dependence. if dependent, find the relation between them.
	
	a. $X_1=(1,-1,1),X_2=(2,1,1),X_3=(3,0,2)$
	
	b. $X_1=(1,2,3),X_2=(2,-2,6)$
	
	c. $X_1=(3,1,-4),X_2=(2,2,-3),X_3=(0,-4,1)$
	
	d. $X_1=(1,1,1,3),X_2=(1,2,3,4),X_3=(2,3,4,7)$
	
	e. $X_1=(1,1,-1,1),X_2=(1,-1,2,-1),X_3=(3,1,0,1)$
	
	f. $X_1=(1,-1,2,0),X_2=(2,1,1,1),X_3=(3,-1,2,-1),X_4=(3,0,3,1)$
	
\item  Show that the colume vectors of following matrix a are linearly independent:
\[
A=
\left[
\begin{array}{rrrr}
	  1  &0&0 \\
	 6 & 2 &1 \\
	 4&3&2\\
\end{array}
\right]
\]
\item  Show that the vectors $X_1=(2,3,1,-1).X_2=(2,3,1,-2),X_3=(4,6,2,1)$ are linearly dependent. Express one of the 	vectors as linear combination of the others.

\item  Find whether or not the following set of vectors are linearly dependent or indipendent:
 
a. $(1,-2),(2,1),(3,2)$
b. $(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1)$

\item  Show that the vectors \[X_1=(a_1,b_2),X_2=(a_2,b_2)\]are linearly dependent if \[a_1~b_2-a_2~b_1=0.\]
 
\end{enumerate}
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